I was going through Neukirch's algebraic number theory again (after years of not studying number theory), and found out that I marked a star on exercise 1.8.2. I tried to solve it again, but I had no idea where to start. The question is as follows:
For every integral ideal $\mathfrak{A}$ of $\mathcal{O}$, there exists a $\theta \in \mathcal O$ such that the conductor $\mathfrak F = \{ \alpha \in \mathcal O : \alpha \mathcal O \subseteq \mathcal o [\theta] \}$ is prime to $\mathfrak A$ and such that $L=K(\theta)$.
Notations : $\mathcal o$ is a Dedekind domain with field of fractions $K$, and $\mathcal O$ is the integral closure of $\mathcal o$ in a finite field extension $L$ of $K$.
It seems quite clear that this exercise is missing the assumption that the field extension $L/K$ is separable, so I will assume that.
I have no idea where to start -- I believe that the biggest problem is that I have absolutely no intuition about what a conductor is. The book gives few theorems about it, but I do not see what it actually is about.
Can someone please explain about this concept, or give some hints on this question? Thank you.
Edit : it seems that the exercise is wrong, but I still seek to understand what conductors are all about.